ON THE FUNDAMENTAL GROUP OF THE SIERPIŃSKI-GASKET

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1 ON THE FUNDAMENTAL GROUP OF THE SIERPIŃSKI-GASKET S. AKIYAMA, G. DORFER, J. M. THUSWALDNER, AND R. WINKLER Dedicated to Professor Peter Kirschenhofer on the occasion of his 50th birthday Abstract. We give a description of the fundamental group π 1 ( ) of the Sierpiński-gasket. It turns out that this group is isomorphic to a certain subgroup of an inverse limit lim G n formed by the fundamental groups G n of natural approximations of. This subgroup, and with it π 1 ( ), can be described in terms of sequences of words contained in an inverse limit of semigroups. 1. Introduction The present paper is devoted to the description of the fundamental group of the Sierpiński-gasket (see Figure 1). It turns out that this fundamental group can be viewed as a subset of an inverse limit of the fundamental groups of certain natural approximations of. Before we give more details we would like to state some definitions and earlier results that are related to our topic. Figure 1. The Sierpiński-gasket Date: March 22, Mathematics Subject Classification. 14F35,28A80. Key words and phrases. Sierpiński-gasket, fundamental group. The authors thank the Austrian Science Foundation FWF for its support through Projects no. S8307, S8312, S8310, P17557-N12, S9610-N13 and S9612-N13. The second author gratefully acknowledges hospitality and support of the National University of Singapore offered during a research visit in August-September

2 2 S. AKIYAMA, G. DORFER, J. M. THUSWALDNER, AND R. WINKLER One of the possibilities to define the Sierpiński-gasket is to use a so-called iterated function system. Let f 1 (x) := x 2, f 2(x) := x , f 3(x) := x Then it is well-known that C is the unique non-empty compact subset of C satisfying the set equation 3 = f j ( ) j=1 (see for instance Hutchinson [16]). Since f 1, f 2, and f 3 are similarities, is a selfsimilar set. Topological properties of self-similar sets have been studied extensively in the literature. For instance Hata [13, 14] proves that a connected self-similar set is a locally connected continuum. Moreover, he establishes criteria for the connectivity of a self-similar set, deals with their cut points and proves a criterion for a self-similar set to be homeomorphic to an arc. Cut points play a role also in Winkler [21]. Related questions are addressed by Bandt and Keller [2], where the authors get information on the topological properties of self-similar sets by studying their dynamics. More recently, topological properties of self-similar sets with nonempty interior attracted interest. We mention the survey paper by Akiyama and Thuswaldner [1], where many results are stated. Some results on the structure of the fundamental group of self-similar sets are shown in Luo and Thuswaldner [17]. In describing the fundamental group π( ), the main difficulty consists in the fact that is not locally simply connected. This makes it impossible to apply the classical methods like van Kampen s theorem and the theory of covering spaces in order to compute the fundamental group of. Spaces that are not locally simply connected have been studied for a long time. We want to review some of the known results on such spaces. The standard example of a non-locally simply connected space is the so-called Hawaiian Earring (see Figure 2) which is defined by H := n 1 { z C : z 1 n = 1 n It is not locally simply connected in the origin. Properties of the fundamental group }. Figure 2. The Hawaiian Earring of H were studied by Higman [15]. Morgan and Morrisson [19] determine π(h) as

3 ON THE FUNDAMENTAL GROUP OF THE SIERPIŃSKI-GASKET 3 a subgroup of an inverse limit of finite free products. Their proof was simplified by de Smith [8] who also showed that π(h) is uncountable and not free. More recently, in a big project consisting of three papers, Cannon and Conner [3, 4, 5] thoroughly study fundamental groups (and so-called big fundamental groups) of one-dimensional spaces. In particular, [3] is devoted to the fundamental group of the Hawaiian Earring. The authors give a combinatorial description of this group in terms of big words. In [5] they prove some important properties of the fundamental groups of one-dimensional spaces. For instance, they show that for a one-dimensional space X the following assertions are equivalent: π(x) is free, π(x) is countable, X has a universal cover, X is locally simply connected. Conner and Lamoreaux [7] generalize some of the results of [5] to larger classes of subsets of the plane. In the present paper we embed the fundamental group of into an inverse limit of groups which is easily seen to be equal to the Čech homotopy group of. The fact that the fundamental group of a one-dimensional space is always isomorphic to a subgroup of its Čech homotopy group is proved by Eda and Kawamura [10]. In Eda [9] criteria for the isomorphy of the fundamental group of two non-locally simply connected spaces is studied. More recently, Conner and Eda [6] proved that certain spaces can be recovered from their fundamental groups, the Sierpińskigasket is among these spaces. Finally, we mention that Homology groups of nonlocally simply connected spaces are studied in [11, 12]. The starting point of the present paper is a remark contained in [5, Section 2]. In an example the authors describe the implications of their results for the fundamental groups of Sierpiński and Menger curves. Among other things, they showed that these spaces have uncountable fundamental groups which are not free and that they do not have an universal cover. On the other hand, the authors mention that these groups have no known combinatorial (word) structure. In the present paper we want to describe the fundamental group of the Sierpiński-gasket by some word structure. In what follows we want to give a short overview of the content of the present paper. It is an evident idea to consider for a loop f in the sequence of homotopy classes [f] n of f in the approximating spaces n that arise when the usual construction process of recursively removing the open middle triangle is stopped at level n. Applying the result of Eda and Kawamura [10] mentioned above we can show that the sequence ([f] n ) n 0 characterizes f exactly up to homotopy. The natural ambient space for the sequences ([f] n ) n 0 is the inverse limit lim G n of the fundamental groups G n of n. With an easy example (see Example 2.13) it becomes clear that lim G n contains elements which do not represent homotopy classes for loops in. So the objective appears to describe the subgroup of lim G n that corresponds to the fundamental group of. Our approach to this task pursues the following strategy: Instead of investigating the problem directly in lim G n we consider an intermediate semigroup structure lim S n in which the set S( ) of all loops in is described up to re-parametrization (see Figure 3). To this end at every approximation level n we represent a loop f by a (finite) word σ n (f) consisting of the sequence of transition points (later called dyadic points) between the subtriangles of n that the loop passes. An appropriate reduction process on σ n (f) leads then to a canonical representative of the homotopy class [f] n which as a byproduct gives rise to an adequate representation of the elements in lim G n.

4 4 S. AKIYAMA, G. DORFER, J. M. THUSWALDNER, AND R. WINKLER S( ) [. ] π( ) σ ϕ lim S n Red lim G n Figure 3. We finally succeed in characterizing the elements of the fundamental group of by a, after all, surprisingly simple stabilizing condition in the inverse semigroup limit lim S n. The crucial step towards this result is the fact that though σ is not surjective, restricting the domain of the reduction map Red : lim S n lim G n to Im(σ) does not affect its image, i.e., Im(Red σ) = Im(Red). Moreover, we employ considerable effort to completely describe the kernel and the range of σ to enlighten the relevance of lim S n independently of its expedience with respect to the description of the fundamental group of : The elements in the range of σ are characterized by a completeness condition and they precisely describe the set of all loops in up to re-parametrization. The organization of the two forthcoming chapters is as follows: In Section 2.1 we introduce a digital representation for the points of the Sierpiński-gasket by retracing the usual construction process of recursively removing the open middle triangle. Thereby we obtain two sequences of approximating spaces to, and the points in naturally split into the two classes of dyadic and generic points. In Section 2.2 it is explicated how a loop in can be represented by a finite word over the alphabet of dyadic points of order n at every approximation level n. In Section 2.3 we introduce the inverse limit of semigroups lim S n and show that the semigroup S( ) of all loops in can be mapped by a homomorphism into lim S n by means of the sequence of representations of a loop attained in Section 2.2. In Section 2.4 we introduce the set of reduced words G n which turns out to be isomorphic to the fundamental group of n. The G n, n 0, form an inverse limit of groups lim G n and an appropriate reduction map on elements of lim S n is defined such that the diagram in Figure 3 commutes. Employing a result of Eda and Kawamura [10] we see that ϕ is injective and thus the fundamental group of is a subgroup of lim G n. Example 2.13 demonstrates that ϕ is not surjective. This provided the initial motivation for considering lim S n. Section 2.5 contains the calculation of the Čech homotopy group of. In Section 3.1 we develop the machinery to study the range and the kernel of σ which is accomplished in Propositions in full detail. In Section 3.2 we finally present a characterization of the elements in lim G n representing a homotopy class in π( ). 2. Preliminaries 2.1. Digital representations of the Sierpiński-gasket. For our purposes we need a digital representation of the points of the Sierpiński-gasket. To this end we follow the construction process of that recursively removes the open middle triangle at each stage. We start with a triangle (including its inside) 0 in the plane. Just to have a concrete metric at hand we assume that 0 is equilateral with side length 1. The vertices of 0 are denoted by 0, 1 and 2. By joining the midpoints of the sides 0 is subdivided in four smaller triangles 0, 1, 2 and the

5 ON THE FUNDAMENTAL GROUP OF THE SIERPIŃSKI-GASKET 5 middle triangle, where i is the subtriangle that contains the vertex i. Removing the interior of the middle triangle from 0 we obtain the first approximation 1, i.e., 1 = With the remaining triangles i, i = 0, 1, 2, we proceed in the same way: i is divided into the four subtriangles i0, i1, i2, and the middle triangle the interior of which is cut out in the next step. Thus we get the second approximation 2 = ij, i,j {0,1,2} and so on and so forth. We obtain a decreasing sequence of compact spaces and hence the intersection = n, the Sierpiński-gasket, is a compact space as well. consists of two types of points which we call dyadic and generic: Dyadic points: these are points P which lie in two different subtriangles at some stage (and consequently in all the following stages) in the construction process described before. The smallest level at which P appears as a vertex of two different subtriangles is called the order of P. For instance {P } = = =... defines a point P of order 2. We represent P by (0, 1/2) or (0, 2/1) (see Figue 4). In general a dyadic point of order n has a finite representation of the form P = (a 1, a 2,..., a n 1, a/b) = (a 1, a 2,..., a n 1, b/a) with a i, a, b {0, 1, 2} and a b, and this means {P } = a 1 a 2... a n 1 a a 1 a 2... a n 1 b. We consider the vertices 0, 1, 2 of 0 as dyadic points of order 0. Let in the following D n denote the set of all dyadic points of order n. In D n there is a natural relation n describing the neighborhood of dyadic points at level n: for P, Q D n we have P n Q if and only if P Q and there is a subtriangle a 1... a n of n to which P and Q belong. At every stage n a dyadic point P D n, P 0, 1, 2 has exactly four neighbors, and the points 0, 1 and 2 have exactly two neighbors each. n N 2 (2, 0/2) (2, 1/2) (0/2) (1/2) (0, 0/2) (1, 1/2) 0 (0, 0/1) (0/1) (1, 0/1) 1 Figure 4 Generic points: these are points P of such that at every stage n there is a unique subtriangle of n to which the point P belongs. If P a 1 a 2... a n, n N, then P has the infinite representation P = (a 1, a 2,...) with a i {0, 1, 2}, where the sequence (a n ) n N is not ultimately constant.

6 6 S. AKIYAMA, G. DORFER, J. M. THUSWALDNER, AND R. WINKLER Formally can be obtained as the quotient space of the compact space X of onesided infinite sequences over the three letter alphabet {0, 1, 2}, i.e., X = {0, 1, 2} N with the discrete topology on the factors, where a pair of sequences (a n ) n N and (b n ) n N is identified if there is an n 0 such that a n = b n for n < n 0 and a n = b n0 a n0 = b n for n > n 0. In the approach described before this means that P = (a 1, a 2,..., a n0 1, a n0 /b n0 ) is a dyadic point of order n 0. The spaces n, n 0, provide an encasing approximation to the Sierpińskigasket. In the following we will also consider an approximation from inside. Let n denote the boundary of n considered as a subspace of the plane. Then = n where the bar means the closure operator in the plane: n contains n N exactly those points P = (a n ) such that eventually the digits a n are out of a twoelement subset of {0, 1, 2}, in particular this set contains all dyadic points. On the other hand every generic point of is the limit of a sequence of dyadic points. Concerning homotopy the spaces n and n 1, n 1, provide the same level of approximation to the Sierpiński-gasket. There exists a deformation p n that retracts n to n 1 : For every subtriangle T = a 1 a 2... a n 1 of n 1 the map p n projects the points of n T from the center of T to the boundary of T. Hence the fundamental groups π( n ) and π( n 1 ) are isomorphic (cf. [20, Theorem 1.22 and Theorem 3.10]) Representation of loops in. To describe the fundamental group π( ) we have to consider continuous loops f : [0, 1]. Since is path connected throughout we may assume f(0) = f(1) = 0. Our next aim is to represent f by a finite word over the alphabet D n for every n 0. Let us fix n. The pre-images {f 1 (P ) P D n } form a finite family of disjoint compact subsets of the interval [0, 1]. Therefore this family is separated, i.e., there is m N such that for all i = 1, 2,..., m the set f 1 (P ) [ i 1 m, i m ] is non-empty for at most one P. We list these points P as i increases and in the arising sequence we cancel out consecutive repetitions. Thus we obtain a finite word P 1 P 2... P k =: σ n (f) over D n which is independent of the chosen m and represents f at approximation level n. Obviously σ n (f) has the following properties: (2.1) (2.2) P 1 = P k = 0, P i n P i+1 for all i = 1,... k 1. In the following we will also consider the loop that emerges from σ n (f) by connecting the listed points straight-lined in the order they appear and call it the piecewise linear loop corresponding to σ n (f). In order to disburden the notation we will not distinguish between the string σ n (f) and the associated loop as long as no confusion can arise. Proposition 2.1. In n the loop f and the piecewise linear loop σ n (f) are homotopic. Proof. Let σ n (f) = P 1... P k. For every i = 1,..., k there is a maximal interval [s i, t i ] such that f(s i ) = f(t i ) = P i, f([s i, t i ]) D n = {P i } and 0 = s 1 t 1 < s 2 t 2 <... < s k t k = 1. This means that f([s i, t i ]) is contained in the interior as a subset of n of the union of the (at most) two subtriangles of n that intersect in P i. Since this set is simply connected f [s i, t i ] is homotopic to the constant loop in P i. Moreover, the conditions on s i and t i imply that f([t i, s i+1 ]) is a subset of the subtriangle of n that contains P i and P i+1 and hence f [t i, s i+1 ] is homotopic to the straight line between P i and P i+1. Putting the pieces together we obtain the assertion. n N

7 ON THE FUNDAMENTAL GROUP OF THE SIERPIŃSKI-GASKET 7 In order to describe the fundamental group of, Proposition 2.1 suggests to represent a loop f, as a first step, by the sequence (σ n (f)) n 0. In the next section we will elaborate an appropriate ambient space for the sequences (σ n (f)) n The inverse system (S n, γ n ) n 0 of semigroups. The semigroups S n, n 0, are defined in the following way: The elements of S n are finite words ω n = P 1... P k over the alphabet D n such that (2.1) and (2.2) are satisfied. These words ω n are called admissible and they are supposed to represent paths in n. (2.1) means that we consider only cyclic paths with base point 0, and (2.2) reflects that with respect to homotopy constant parts of paths do not matter and that in a continuous path a dyadic point can only be followed by a neighboring dyadic point. The semigroup operation on S n is defined by concatenation of words and cancellation of one of the adjacent letters 0 at the interface: P 1... P k Q 1... Q l = P 1... P k Q 2... Q l. The bonding mapping γ n : S n S n 1, n 1, eliminates from an element of S n all points of order n, and then cancels consecutive repetitions of points of order < n arising in this process. Obviously the result is an admissible word in S n 1 and γ n is a semigroup epimorphism. Thus we may consider the inverse semigroup-limit lim S n = {(ω n ) n 0 γ k (ω k ) = ω k 1 for all k 1} corresponding to the sequence (S n, γ n ) n 0. Let (S( ), ) denote the semigroup of continuous loops f : [0, 1] where multiplication is just the usual concatenation of loops. As a general principle we denote the semigroup operations in S( ), S n and lim S n by (or omit the operation symbol), whereas for the group operations, for instance in the fundamental group π( ), we use the notation. Next we will provide a digital description of loops at the semigroup level. Proposition 2.2. The map σ : is a semigroup homomorphism. { S( ) lim S n f (σ n (f)) n 0 Proof. Firstly we show that σ is well defined: Let f be an element of S( ). Then the word σ n (f) contains the dyadic points of D n which are passed by the loop f in the order they appear in f without consecutive repetitions. When we apply γ n to σ n (f), obviously we end up with the same word in S n 1 we obtain when we list the dyadic points f passes at level n 1, i.e., γ n (σ n (f)) = σ n 1 (f), and thus σ(f) lim S n. σ is a homomorphism since concatenation of loops in S( ) correlates exactly to the concatenation of words in the components S n, n 0. To put it more formally, for f, g S( ) we have: σ(f g) = (σ n (f g)) n 0 = (σ n (f) σ n (g)) n 0 = (σ n (f)) n 0 (σ n (g)) n 0 = σ(f) σ(g) The inverse system (G n, δ n ) n 0 of groups. In order to describe the homotopy of loops in we have to consider an appropriate reduction process for the semigroup words in lim S n. In the following for f S( ) let [f] denote the homotopy class of f in, and let [f] n denote the homotopy class of f in n, i.e., in the latter case f is considered as a map with range n.

8 8 S. AKIYAMA, G. DORFER, J. M. THUSWALDNER, AND R. WINKLER In a first step we will describe the elements of the fundamental group of n. Very briefly we recall here the standard approach to the fundamental group of a simplicial complex (cf. [20, chapter 7]): One considers edge paths in n which start and end in the same vertex, say in 0. In principle an edge path is the same as an admissible word over D n, i.e., an element of S n, except that also constant edges are allowed. Two edge paths are defined to be equivalent if one can be obtained from the other by a finite number of elementary moves. In our setting an elementary move is a substitution on subwords consisting of consecutive letters of the form (2.3) P QP P or P QR P R where P, Q, R are the distinct vertices of a simplex in the simplicial complex which in our case means that P, Q, R form a subtriangle of n. As the arrows indicate these transformations may be performed in both directions. The equivalence classes of edge paths then constitute the elements of the fundamental group with concatenation as the group operation (cf. [20, Theorem 7.36]). In our attempt we proceed slightly different: We call an element ω n S n reduced if ω n cannot be shortened by an elementary move as described in (2.3). A reduced word in S n can be identified with a sequence of subtriangles of n such that any three consecutive subtriangles are pairwise different. Let G n denote the set of all reduced words of S n and Red n : S n G n the mapping that performs elementary moves until the word is reduced. Proposition 2.3. Red n is well defined and for ω n S n the loop corresponding to Red n (ω n ) forms a canonical representative of the homotopy class of the loop corresponding to ω n in n. Proof. Obviously, by performing an elementary move on an element of S n we stay in the same homotopy class for the corresponding loops. All we have to show is that two different reduced words correspond to non-homotopic loops. Here we use the fact that n and n 1 have isomorphic homotopy groups ( n 1 is a deformation retract of n ). Since n 1 is a connected 1-complex its homotopy group is a free group, freely generated by the edges not contained in a fixed spanning tree T (cf. [20, Corollary 7.35]). Starting with two different reduced words ω n ω n in G n by retracting the loops corresponding to ω n and ω n to n 1, we end up with two different words ω n 1 ω n 1 over the alphabet D n 1 such that any three consecutive letters of these words are pairwise different elements of D n 1 (a reduced word in G n correlates to a sequence of subtriangles in n ; every subtriangle in n contains exactly one vertex in D n 1 ; the sequence of these vertices is exactly what we obtain by the retraction). Suppose the two emerging loops corresponding to ω n 1 and ω n 1 are homotopic in n 1, then due to the fact that the homotopy group of n 1 is a free group the two words must contain the same edges not contained in the tree T in the corresponding order. Moreover, there is a unique path in T connecting these edges. Since ω n 1 and ω n 1 do not contain subwords of the form P QP, ω n 1 and ω n 1 must be identical in the parts connecting the edges not in T, and hence they must coincide on the whole, which is a contradiction. Now it is obvious how to define the group operation for ω n, ω n G n : ω n ω n = Red n (ω n ω n ), where ω n ω n is the product in S n. Together with the results in [20, chapter 7] we obtain:

9 ON THE FUNDAMENTAL GROUP OF THE SIERPIŃSKI-GASKET 9 Proposition 2.4. (G n, ) is isomorphic to the fundamental group (π( n ), ) by means of the isomorphism ϕ n : [f] n Red n (σ n (f)) where f is a continuous loop in n. Furthermore, the reduction map Red n : S n G n, associating to every admissible word its reduced form, is a semigroup epimorphism, i.e., (G n, ) is isomorphic to (S n / ker(red n ), ). Now we elaborate a bonding between the groups G n. Lemma 2.5. For n 1 the map { Gn G δ n : n 1 ω n Red n 1 (γ n (ω n )) is a group epimorphism. Proof. Let ω n, ω n G n. We have On the other hand we get δ n (ω n ω n ) = Red n 1 (γ n (Red n (ω n ω n ))). δ n (ω n ) δ n ( ω n ) = Red n 1 (Red n 1 (γ n (ω n )) Red n 1 (γ n ( ω n ))) = Red n 1 (γ n (ω n ) γ n ( ω n )) = Red n 1 (γ n (ω n ω n )). Due to Proposition 2.3 it is thus sufficient to show that the loops γ n (Red n (ω n ω n )) and γ n (ω n ω n ) are homotopic in n 1. It is obvious by the definition of γ n that [α n ] n 1 = [γ n (α n )] n 1 for every α n S n. Further we have [α n ] n = [Red n (α n )] n and hence also [α n ] n 1 = [Red n (α n )] n 1. Altogether we obtain [γ n (ω n ω n )] n 1 = [ω n ω n ] n 1 = [Red n (ω n ω n )] n 1 = [γ n (Red n (ω n ω n ))] n 1 and we are done. δ n is surjective: Suppose ω n 1 = P 1 P 2... P k in G n 1 is given. Put ω n = P 1 Q 1 P 2 Q 2... Q k 1 P k, where Q i is the (unique) element of D n with P i n Q i n P i+1. One can check easily that ω n is reduced and δ n (ω n ) = ω n 1. As a consequence of the last lemma we can consider the inverse group-limit lim G n = {(ω n ) n 0 δ k (ω k ) = ω k 1 for all k 1}. Next we show that the reduction maps Red n : S n G n can be lifted to a map on the inverse limits. Lemma 2.6. For every n 1 the following diagram commutes: γ n S n Sn 1 Red n Red n 1 δ n Gn 1 G n Proof. Let ω n be in S n. We have to show that δ n (Red n (ω n )) = Red n 1 (γ n (ω n )). Since δ n (Red n (ω n )) = Red n 1 (γ n (Red n (ω n ))) it suffices to prove that γ n (ω n ) and γ n (Red n (ω n )) are homotopic in n 1. However, this was already accomplished in the proof of Lemma 2.5. Proposition 2.7. The map { lim S n lim G n Red : (ω n ) n 0 (Red n (ω n )) n 0 is a well defined semigroup homomorphism.

10 10 S. AKIYAMA, G. DORFER, J. M. THUSWALDNER, AND R. WINKLER Proof. If (ω n ) n 0 lim S n then γ n (ω n ) = ω n 1 for every n 1. This yields δ n (Red n (ω n )) = Red n 1 (γ n (Red n (ω n ))) = Red n 1 (γ n (ω n )) = Red n 1 (ω n 1 ) where the penultimate identity was derived in the proof of Lemma 2.6. This shows that Red is well defined. Utilizing Proposition 2.3 in a similar way it is straightforward to prove that Red is a homomorphism. Now we figure out that the fundamental group (π( ), ) can be embedded into the group-limit (lim G n, ). Proposition 2.8. The map ϕ : is a well defined group homomorphism. { π( ) lim G n [f] Red(σ(f)) Proof. Let f, g be continuous loops in. We recall (Proposition 2.4) that Red n (σ n (f)), considered as a piecewise linear path, is the canonical representative of [f] n, the homotopy class of f in n, n 0. If [f] = [g] then, of course, [f] n = [g] n for all n. But this means Red n (σ n (f)) = Red n (σ n (g)) for all n and hence Red(σ(f)) = Red(σ(g)). This shows that ϕ is well defined. With the results we already have it is straightforward to prove that ϕ is a homomorphism Čech homotopy of and injectivity of ϕ. In a next step we want to prove the injectivity of ϕ. To this matter we first construct the Čech homotopy group ˇπ( ) of (see e.g. [18, p. 130] 1 or [10, Appendix A] for a definition of ˇπ). Since it will turn out that ˇπ( ) = lim G n, the injectivity of ϕ follows from the fact that the fundamental group of a one-dimensional space can be embedded in its Čech homotopy group (cf. [10, Theorem 1.1]) and ϕ is the corresponding canonical embedding. Before we give the details we have to set up some notation. Note that each of the equilateral triangles T = a 1 a 2... a n has side length 1 2. n For a small ε > 0 let U T be the open equilateral triangle of side length (1 + ε) 1 2 n having the same midpoint and parallel sides to T. For each n 0 define the collections U n := {U T T = a 1... a n with a 1,..., a n {0, 1, 2}}. Now choose ε in a way that U n is a cover of order 2 for each n (i.e. for all pairwise distinct sets U, U, U U n we have U U U = ). This implies that (2.4) U a1...a n U a 1...a n a 1... a n a 1... a n =. Employing compactness of we know that the Čech homotopy group can be defined in terms of finite covers. U n is a finite, open cover of n and thus also of. Moreover, it is easy to see that (U n ) n 0 is a sequence of covers of which is cofinal in the set of all finite open covers of. Thus, denoting the nerve of a cover C by N (C), we have (2.5) ˇπ( ) = lim π(n (U n )). Now we are in a position to prove that the nerve N (U n ) is homotopy equivalent to n for each n 1. (In what follows, homotopy equivalence will be denoted by.) 1 Note that the Čech homotopy group is called shape group in this text.

11 ON THE FUNDAMENTAL GROUP OF THE SIERPIŃSKI-GASKET 11 Lemma 2.9. For each n 1 we have N (U n ) n 1 n. Proof. We start with proving the first homotopy equivalence by induction on n. For n = 1 the sets N (U n ) and n 1 are both homeomorphic to a circle and thus homotopy equivalent. Assume that the result is proved for a certain n. Now we are going to construct the nerve of U n+1. Consider the subdivision with U n+1 = U (0) n+1 U (1) n+1 U (2) n+1 U (i) n+1 := {U T T = ia 2... a n+1 with a 2,..., a n+1 {0, 1, 2}}. Then N (U (i) n+1 ) is homeomorphic to N (U n). Thus N (U n+1 ) contains three copies of N (U n ). Each of these copies is homotopy equivalent to n 1 by induction (see Figure 5 where the situation is depicted for n + 1 = 3). Figure 5. On the left hand side the covers U (i) 3 for i {0, 1, 2} are depicted separately. Right you can see (deformation retracts of) the associated nerves N (U (i) 3 ). Now we have to determine the overlaps between the elements of the covers U (i) n+1. Let U T, U T be two elements of U n+1 having non-empty intersection. If both of these sets are contained in U (i) n+1 for the same i {0, 1, 2} the 1-simplex caused by this intersection in N (U n+1 ) is already contained in N (U (i) n+1 ). Figure 6. On the left hand side we illustrate the intersections between the covers U (i) 3 (0 i 2). The pairs of black triangles connected by a line segment intersect in U 3. This leads (see right) to additional 1-simplices between the nerves of U (i) 3. Suppose on the other hand that U T U (i) n+1 and U T U (i ) n+1 for i i. From (2.4) we see that this implies that one of the following three constellations holds

12 12 S. AKIYAMA, G. DORFER, J. M. THUSWALDNER, AND R. WINKLER (C.1) T = and T = (or vice versa), (C.2) T = and T = (or vice versa), (C.3) T = and T = (or vice versa). The constellation in (C.i) gives rise to a 1-simplex leading from N (U (i 1) n+1 ) to (i mod 3) N (U n+1 ) in N (U n+1 ) (see Figure 6 for an illustration in the case n + 1 = 3). Summing up we have shown that N (U n+1 ) is the union of three homotopic copies of N (U n ) n 1 connected cyclically by 1-simplices. By shrinking these three 1- simplices to points it is easy to see that N (U n+1 ) deformation retracts to n 1 and the first assertion is proved. The second assertion was already verified in the last paragraph of Section 2.1. Lemma The Čech homoptopy group ˇπ( ) is isomorphic to lim G n. Proof. This follows by combining Lemma 2.9 with (2.5) and Proposition 2.4. Proposition The group homomorphism ϕ defined in Proposition 2.8 is injective. Proof. Composing ϕ : π( ) lim G n with the canonical isomorphism between lim G n and ˇπ( ) (see Lemma 2.10) we obtain the canonical homomorphism from π( ) to ˇπ( ). Since is a one-dimensional continuum, [10, Corollary 1.2] implies that this canonical homomorphism is injective and so is ϕ. The next theorem gives an interim survey of what we have established up to this point. Theorem The fundamental group (π( ), ) of the Sierpiński-gasket is isomorphic to a subgroup of (lim G n, ). Moreover, the following diagram commutes: S( ) [. ] π( ) σ ϕ lim S n Red lim G n However, the next example shows that ϕ is not surjective: Example Let C 0 be the (piecewise linear) loop that starting at 0 passes around the boundary of 0 in positive direction (i.e. passing from 0 to 1, then 2 and back to 0). By C0 1 we mean the same cycle passed in the opposite direction. C 1 denotes the loop around the subtriangle 0 in 1 (i.e. passing through 0, (0/1), (0/2) and 0), C 2 the loop around 00 in 2, and so on. Now we consider the following sequence of words: ω 0 = ω 1 = 0 ω 2 = Red 2 (σ 2 (C 0 C 1 C 1 0 )) ω 3 = Red 3 (σ 3 (C 0 C 1 C 1 0 C 2)) ω 4 = Red 4 (σ 4 (C 0 C 1 C 1 0 C 2C 0 C 3 C 1 0 )) ω 5 = Red 5 (σ 5 (C 0 C 1 C 1 0 C 2C 0 C 3 C 1 0 C 4))... It can be checked easily that (ω n ) n 0 is an element of lim G n. For instance, if we apply δ 4 to ω 4, the loop C 3 disappears since it is null-homotopic in 3, and consequently also the C 0 and C0 1 neighboring C 3 cancel out and we arrive at ω 3.

13 ON THE FUNDAMENTAL GROUP OF THE SIERPIŃSKI-GASKET 13 Suppose there exists f in S( ) such that ϕ([f]) = (ω n ) n 0. Then due to the construction of ω n = [f] n the loop f has to traverse the circle C 0 infinitely many times, which is not possible. Maybe it is instructive to see here that (ω n ) n 0 is even not in Red(lim S n ). Suppose there is (α n ) n 0 in lim S n with Red((α n ) n 0 ) = (ω n ) n 0. If we consider only the dyadic points of order 1 that appear in ω 2n, we see that the sequence (0/1) (1/2) (0/2) (1/2) (0/2) repeats n times. This means that at least this sequence of 5n points of order 1 also appears in α 2n (maybe some more which cancel out by performing Red 2n ). However, when projecting down from S 2n to S 1 in lim S n no cancellation in between this 5n points can occur. As a consequence α 1 would contain infinitely many points which is a contradiction. We aim at describing the fundamental group of the Sierpiński-gasket. Retrospectively, Theorem 2.12 provides the motivation for investigating the semigroup limit lim S n : π( ) = ϕ(π( )) = Red(σ(S( ))). Therefore we have to study the range of σ in lim S n and the range of Red in lim G n. This will be accomplished in the next section. 3. A characterization of the elements in ϕ(π( )) 3.1. The range and the kernel of σ. We associate to a fixed element (ω n ) n 0 = (P n1 P n2... P nkn ) n 0 in lim S n a graph G = (V, E) with vertices V and directed edges E. We think of the graph G as organized in rows: in the nth row, n 0, we have for every letter appearing in the word ω n a corresponding vertex, i.e. V = {(n, j) n 0, 1 j k n }. Edges connect certain vertices from row n to vertices in row n + 1, namely, ((n, i), (n + 1, j)) E if and only if P ni = P n+1,j and in the course of γ n+1 that maps ω n+1 to ω n the point P n+1,j is projected to P ni. Consequently any vertex (n, i) in row n has at least one successor up to a finite number of successors (not bounded from above for growing n) in row n + 1, and (n, i) has exactly one predecessor in row n 1 if and only if the order of P ni is < n. Example 3.1. We consider the following element in lim S n one can think of as a pseudo-path that passes from 0 on the baseline of 0 arbitrarily near to 1 without touching 1 and then goes the same way back to 0. A phenomenon arising in this example will turn out to be important in the further investigation: ω 0 = 0, ω 1 = 0(0/1)0, ω 2 = 0(0, 0/1)(0/1)(1, 0/1)(0/1)(0, 0/1)0,.... Figure 7 shows the graph associated to (ω n ) n 0 where we denote the vertices by the corresponding dyadic points P ni instead of the index (n, i) we usually use. 0 0 (0/1) 0 0 (0, 0/1) (0/1) (1, 0/1) (0/1) (0, 0/1) 0 Figure 7

14 14 S. AKIYAMA, G. DORFER, J. M. THUSWALDNER, AND R. WINKLER By a branch B we mean a directed path in G which cannot be extended. As description for B we use the sequence of vertices contained in B, i.e. B = (n, i n ) n n0 where P = P n,in for all n n 0, is a point of order n 0. We say that the branch B corresponds to the dyadic point P. The set B of all branches in G carries a natural total order : Let B 1 = (n, i n ) n n1, B 2 = (n, j n ) n n2 be two branches then we define B 1 < B 2 if and only if there exists n max{n 1, n 2 } such that i n < j n. Consequently we then have i m < j m for all m n, and i m j m for all m with max{n 1, n 2 } m < n which reflects the property that branches do not cross in G if we display the vertices in every row n in the order they appear in ω n. It is straightforward to check that is a total order on B. For instance, B 1 B 2 and B 2 B 1 implies B 1 = B 2 since branches are maximal with respect to extension. The order on B is dense: Let B 1 < B 2 be defined as before with i n < j n. Then j n+1 i n+1 2 since the points corresponding to B 1 and B 2 are of order n and thus P n+1,in+1 n+1 P n+1,jn+1. Hence any branch B starting at vertex (n + 1, i n+1 + 1) satisfies B 1 < B < B 2. In the following we will consider Dedekind cuts in (B, ): A cut (B 1, B 2 ) is a partition of B into two (nonempty) subsets B 1 and B 2 such that B B 1, B < B implies B B 1, and B B 2, B > B implies B B2. The cut (B 1, B 2 ) is called rational if either B 1 has a largest element or B 2 has a least element. In the remaining case (B 1, B 2 ) is called irrational. Every cut (B 1, B 2 ) converges to a uniquely defined element of in the following sense: For all n 0 put l n = max{i B B 1 : B contains (n, i)} r n = min{j B B 2 : B contains (n, j)} Obviously we have 1 l n r n k n for all n 0. Lemma 3.2. For the cut (B 1, B 2 ) we have lim P n,l n = lim P n,r n. n n Proof. By construction of l n and r n we have either l n = r n and thus P n,ln = P n,rn or r n = l n + 1 and thus P n,ln n P n,rn. Hence it is sufficient to prove the existence of lim P n,l n n. We prove now for all n 0 that P n+1,ln+1 lies in the same subtriangle T n of n as P n,ln : We suppose P n,ln n P n,rn, the other case P n,ln = P n,rn is proved similarly. Let B 1 = (..., (n, l n ), (n + 1, i),...) be a branch in B 1 such that i is a large as possible. Further, let B 2 = (..., (n, r n ), (n + 1, j),...) be a branch in B 2 such that j is a small as possible. Note that P n+1,i = P n,ln, P n+1,j = P n,rn and l n+1 i. Evidently, all points P n+1,k with i < k < j are of order n + 1 and lie in the same subtriangle T n of n as P n,ln and P n,rn, and it is clear by construction that P n+1,ln+1 is one of the points P n+1,k or coincides with P n,ln. Thus we obtain a sequence of subtriangles (T n ) n 0 with T n T n+1, diam(t n ) = 2 n, P n,ln T n, and hence lim P n,l n n exists. The limit of the cut (B 1, B 2 ) is defined to be the point lim P n,l n = lim P n,r n n n in. As the proof of Lemma 3.2 shows, a rational cut has a dyadic limit point, namely the point corresponding to the largest branch in B 1 or the smallest branch in B 2, respectively. An irrational cut may converge to a dyadic or to a generic point. We call (ω n ) n 0 lim S n complete if every irrational cut in the set of branches B associated to (ω n ) n 0 converges to a generic point. Coming back to Example 3.1 we see that (ω n ) n 0 defined there is not complete: Let B 1 consist of all branches which turn left when following them downwards, B 2

15 ON THE FUNDAMENTAL GROUP OF THE SIERPIŃSKI-GASKET 15 all that turn right. Then obviously this cut is irrational and converges to the dyadic point 1. Next we prove that completeness is a necessary condition for (ω n ) n 0 to be an element of σ(s( )). Proposition 3.3. For all f S( ) the representation σ(f) in lim S n is complete. Proof. Put (ω n ) n 0 = (P n1 P n2... P n,kn ) n 0 = (σ n (f)) n 0 and let B = (n, i n ) n n0 be a branch in the graph G which is associated to (ω n ) n 0. We will assign to B an interval [s B, t B ] [0, 1]: Firstly, as we already explicated in the beginning of the proof of Proposition 2.1, for every n n 0 we can associate to P n,in the interval [s n, t n ] such that f([s n, t n ]) D n = {P n,in }. The definition of the edges in the graph G yields [s n+1, t n+1 ] [s n, t n ], and so we obtain a nonempty interval [s B, t B ] = [s n, t n ] such that f is constant on [s B, t B ] with the dyadic n 0 point corresponding to B as the constant value. We list some properties of this relationship between branches and intervals. The order on the branches is preserved by this construction, i.e., if B 1 = (n, i (1) n ) n n1, B 2 = (n, i (2) n ) n n2 are two branches then B 1 < B 2 implies t B1 < s B2 : B 1 < B 2 means that there is an n such that i (1) n < i (2) n and thus for the intervals [s nk, t nk ] associated to P (k) n,i, k = 1, 2, we have t n1 < s n2. Hence n t B1 = inf n n1 t n1 < sup n n2 s n2 = s B2. As a consequence different branches lead to disjoint intervals. Further, it is evident that for every u [0, 1] such that f(u) is a dyadic point there exists a unique branch B with u [s B, t B ]. To sum up, the family {[s B, t B ] B B} forms a partition of f 1 ( D n ) which inherits the order on the set of all branches B in the sense explained above. Now we are in position to prove that every irrational cut (B 1, B 2 ) in B converges to a generic point in : The irrational cut (B 1, B 2 ) corresponds to an irrational cut in {[s B, t B ] B B}. Put s = sup B B1 s B and t = inf B B2 s B. Since the cut is irrational it is irrelevant if we take s B or t B when forming the inf and the sup, and moreover we have s > s B1 and t < t B2 for all B 1 B 1, B 2 B 2. Obviously s t and we claim that f is constant in the interval [s, t] with a generic point as constant value: Suppose there exists u [s, t] such that f(u) is a dyadic point. Then there is a branch B with u [s B, t B]. However, due to the definition of s = sup B B1 s B all intervals corresponding to branches of B 1 are strictly below s and thus cannot contain u. The same applies to all branches of B 2 since their intervals lie above t. Hence B is not in B 1 B 2 = B which is a contradiction. So f does not attain a dyadic point as value on the interval [s, t]. Suppose f is not constant on [s, t]. Then f([s, t]) is a connected subset of containing at least two points and therefore also contains a dyadic point. Finally we show that the cut (B 1, B 2 ) converges to the generic point f(s). Put l n = max{i B B 1 : B contains (n, i)}. Thus for every n n 1 there exists a branch B n = (m, i (n) m ) m mn B 1 such that (n, l n ) = (n, i (n) n ) and thus P n,ln =. As a consequence f(s Bn ) = P n,ln where as usual [s Bn, t Bn ] is the interval P n,i (n) n corresponding to B n. Since B 1 has no largest element for every B = (n, i n ) n n0 B 1 there exists B = (n, j n ) n n0 B 1 with B > B, i.e. there is an n N such that i n < j n l n = i (n) n. This means that for all B B 1 there is an n N such that s B < s Bn. So we infer lim s B n = s, and using the continuity of f we obtain n lim n P n,l n = lim n f(s B n ) = f(s) n 0

16 16 S. AKIYAMA, G. DORFER, J. M. THUSWALDNER, AND R. WINKLER and we are done. We have already seen that non-complete elements in lim S n exist (see Example 3.1). Proposition 3.3 thus shows that σ : S( ) lim S n is not surjective. The next proposition aims at finding f in S( ) such that σ(f) approximates a given (ω n ) n 0 best possible. Proposition 3.4. For every (ω n ) n 0 lim S n there exists f S( ) such that Red(σ(f)) = Red((ω n ) n 0 ). Moreover, if (ω n ) n 0 is complete then even σ(f) = (ω n ) n 0 holds for some f S( ). Proof. Let (ω n ) n 0 = (P n1 P n2... P n,kn ) n 0 be a fixed element of lim S n. We will define a sequence of functions (f n ) n 0 by induction on n such that f n is piecewise linear with range in n and σ k (f n ) = ω k for all k n. We start with n = 0, ω 0 = P 01 P P 0,k0, and divide [0, 1] into 2k 0 1 subintervals of equal length by the points 0 = u 01 < v 01 < u 02 < v 02 <... < u 0,k0 < v 0,k0 = 1. Define f 0 (t) = P 0i for t [u 0i, v 0i ], 1 i k 0, and f 0 to be the linear connection of P 0i and P 0,i+1 in the interval [v 0i, u 0,i+1 ], 1 i < k 0. Obviously σ 0 (f 0 ) = ω 0. Suppose f n is already defined: f n (t) = P ni for t [u ni, v ni ], 1 i k n, f n is the linear connection of P ni and P n,i+1 in the interval [v ni, u n,i+1 ], 1 i < k n, and thus σ k (f n ) = ω k for all k n. We explain in detail how to define f n+1 (t) for t [u n1, v n1 ] and t [v n1, u n2 ]. In the equality γ n+1 (ω n+1 ) = ω n we analyze the action of γ n+1 on the individual letters of ω n+1 : Figure 8 is part of the graph G P n1 P n P n+1,1... P n+1,i1 P n+1,i P n+1,i2 P n+1,i P n+1,i3... Figure 8 we associated to (ω n ) n 0 in the beginning of this section and should be interpreted as follows: P n+1,1 respectively P n+1,i1 is the first respectively last letter in ω n+1 that is projected to P n1 by γ n+1 ; P n+1,i1 +1 up to P n+1,i2 are all of order n + 1 and disappear by applying γ n+1, and so on. Now we define f n+1 (t) for t [u n1, v n1 ] analogously to f 0 in [0, 1]: divide [u n1, v n1 ] into 2i 1 1 subintervals of equal length and define f n+1 in these subintervals alternately to be constant P n+1,i, 1 i i 1, and to connect P n+1,i with P n+1,i+1 linearly, 1 i i 1 1. Next, the interval [v n1, u n2 ] is divided into 2(i 2 i 1 ) + 1 subintervals. Here f n+1 alternately connects P n+1,i with P n+1,i+1 linearly, i 1 i i 2, and is constant P n+1,i, i i i 2. In the same manner we proceed with the rest of the intervals and obtain f n+1 satisfying our requirements. We compare f n with f n+1 (see Figure 9). For 1 i k n : f n (t)... constant P ni t [u ni, v ni ] : f n+1 (t)... stays in the two subtriangles T 1 and T 2 of n that intersect in P ni,

17 ON THE FUNDAMENTAL GROUP OF THE SIERPIŃSKI-GASKET 17 and for 1 i k n 1: f n (t)... connects P ni and P n,i+1 linearly t [v ni, u n,i+1 ] : f n+1 (t)... stays in the subtriangle T 2 of n to which P ni and P n,i+1 belong. Summing up we obtain f n f n+1 2 n where. denotes the maximum norm for t [0, 1]. Consequently f n converges for n uniformly to a continuous f : [0, 1]. By construction we have f m (u ni ) = P ni, 1 i k n, for all m n and thus also f(u ni ) = P ni, 1 i k n. This means that σ n (f) contains at least all letters appearing in the word ω n in the proper order, but it may happen that σ n (f) in between the P ni contains further dyadic points of order n and some of the P ni appear in multiplied form. To illustrate this we consider the interval [u ni, u n,i+1 ]: T 3 R 3 P n,i+1 T 1 T 2 R 1 P ni R 2 Figure 9 f n+1 and all f m with m n + 1 stay for t (u ni, u n,i+1 ) in the interior of the union of the two subtriangles int(t 1 T 2 ) of n (interior as a subset of n ). This implies that f = lim f m stays in the union of the (closed) subtriangles m T 1 T 2. Hence σ n (f [u ni, u n,i+1 ]) = P ni Q 1 Q 2... Q l P n,i+1, l 0, where Q i {R 1, R 2, R 3, P ni, P n,i+1 }. However, since f([u ni, u n,i+1 ]) (T 3 \ {R 3, P n,i+1 }) =, the two letters R 3 and P n,i+1 can never occur in immediate succession in P ni Q 1 Q 2... Q l P n,i+1. This implies that Red n (σ n (f [u ni, u n,i+1 ])) = P ni P n,i+1 and hence on the whole Red n (σ n (f)) = Red n (ω n ). Of course, configurations for P ni and P n,i+1 different to the one displayed in Figure 9 are possible. However, as can be checked easily the consequences concerning the respective subtriangles T 1, T 2 and T 3 are always the same. The first part of the proposition is proved. Now we show that σ n (f) = ω n for all n 0 if (ω n ) n 0 is complete. We have two sets of branches: The set B f corresponding to σ(f) and B ω corresponding to (ω n ) n 0. As pointed out above the vertices of the graph G ω associated to (ω n ) n 0 form a subset of the vertices of the graph G f associated to σ(f). In order to distinguish between these two graphs we use the following notation: Put σ n (f) = (Q n1... Q n, kn ), n 0, and let V f = {(n, j) (f) n 0, 1 j k n } be the vertices in G f. Next it will be outlined that in a canonical way to every branch B = (n, i n ) n n0 in B ω a branch in B f is associated. Two cases may occur: (1) The interval [u, v] = [u n,in, v n,in ] corresponding to B is a singleton. n n 0 Recall that when constructing f n we assigned to every P ni the interval [u ni, v ni ] on which f n has constant value P ni. Thus the property u = v is equivalent to the feature that in G ω for an infinite number of n the vertex

18 18 S. AKIYAMA, G. DORFER, J. M. THUSWALDNER, AND R. WINKLER (n, i n ) has more than one successor: if there is more than one successor of (n, i n ) then [u n+1,in+1, v n+1,in+1 ] has length less than 1/3 of [u n,in, v n,in ]. Let P be the point corresponding to the branch B then in this case f(u) = P and in every neighborhood of u, f has infinitely many different dyadic values. Anyway, turning to the graph G f we see that there is a unique branch B = (n, j n ) (f) n n 0 in B f such that Q n,jn = P corresponds to the interval [s n,jn, t n,jn ] in the sense utilized in the proof of Proposition 2.1 with u [s n,jn, t n,jn ] for all n n 0. (2) The interval [u, v] corresponding to B satisfies u < v. This means that there exists an index n 1 such that for all n n 1 the interval [u n,in, v n,in ] = [u, v]. In this case f n has constant value P on [u, v] for all n n 1 and hence f satisfies this, as well. Again, there exists a unique branch B = (n, j n ) (f) n n 0 in B f such that Q n,jn = P corresponds to the interval [s n,jn, t n,jn ] with [u, v] [s n,jn, t n,jn ]. In the following we will identify B B ω with the respective B B f from (1) or (2) and thus we may consider B ω as a subset of B f. We have already proved in Proposition 3.3 that B f is complete. Now we show that B ω is dense in B f, i.e. for all B 1, B 2 B f with B 1 < B 2 there exists B B ω such that B 1 < B < B 2 : First of all, it is sufficient to prove this for B 1, B 2 B f \B ω : if B 1, B 2 B ω then there exists an according B since is a dense order on B ω, if B 1 B ω, B 2 B f \ B ω, then, since B f is dense, there exists B 3 B f with B 1 < B 3 < B 2 ; if B 3 B ω we are done and if B 3 B f \ B ω then the problem is reduced to B 3 < B 2, the case we will deal with. Let B i correspond to the interval [s i, t i ], f(s i ) = Q i, i = 1, 2. As B 1 < B 2 we have t 1 < s 2. Since B f is dense there exist B 3 B f with B 1 < B 3 < B 2 and since f cannot be constant on [t 1, s 2 ] we can choose B 3 such that the point Q 3 corresponding to B 3 satisfies Q 1 Q 3 Q 2. Consequently there is s 3 (t 1, s 2 ) with f(s 3 ) = Q 3. We fix some k 0 such that the distance d(q 3, Q i ) is larger than 2 k+2, i = 1, 2. Since (f m ) m 0 converges uniformly to f we have f f m < 2 k for all m m k with appropriate m k. So for m m k we have d(q 1, f m (t 1 )) < 2 k, d(q 3, f m (s 3 )) < 2 k. Hence f m (t) must pass from the 2 k -neighborhood of Q 1 for t = t 1 to the 2 k - neighborhood of Q 3 for t = s 3 and since f m is alternately constant/linear f m assumes a dyadic point P (of order m) as constant value for some interval in (s 1, s 3 ). Since σ m (f m ) = ω m there is a branch B B ω containing this P which by construction satisfies B 1 < B < B 3 < B 2. Finally we show σ(f) (ω n ) n 0 (which is equivalent to B f \ B ω ) implies that (ω n ) n 0 is not complete: Let B = (n, i n ) (f) n n 0 B f \ B ω such that for all n n 1 the vertices (n, i n ) (f) in B have smallest possible i n. For instance this is possible if (n 1 1, i n1 1) (f) is a vertex in G f \ G ω. We consider the following cut in B ω : B 1 = {B B ω B < B}, B 2 = {B B ω B > B}. First we show that (B 1, B 2 ) is irrational: for B 1 B 1 we have B 1 < B and since B ω is dense in B f there is B B ω such that B 1 < B < B showing that B 1 has no largest element. Analogously one learns that B 2 has no least element.